A175340 Numbers that can be written as a product of k consecutive composite numbers and also of k+1 consecutive composite numbers, for some k>1, with no factor used twice.

1680, 4320, 120960, 166320, 175560, 215760, 725760, 1080647568000

Offset: 1

Author

Manuel Valdivia, Apr 16 2010, Apr 18 2010

Keywords

• nonn
• more

Comments

The term 725760 has three representations of 4, 5 and 6 numbers (with overlap).

From David A. Corneth, Mar 27 2021, Mar 28 2021: (Start)

a(12) > 10^22 if it exists.

Let x be the product of k consecutive composite numbers and y be the product of k+1 consecutive composite numbers giving some a(m). This sequence does not allow factors of x and y to overlap. If we do allow such overlaps we get A342876. (End)

Examples

			1680 = 10*12*14 = 40*42.
4320 = 6*8*9*10 = 15*16*18.
120960 = 8*9*10*12*14 = 16*18*20*21.
166320 = 18*20*21*22 = 54*55*56.
175560 = 55*56*57 = 418*420.
215760 = 58*60*62 = 464*465.
725760 = 12*14*15*16*18 = 27*28*30*32.
1080647568000 = 49*50*51*52*54*55*56 = 98*99*100*102*104*105.
From _David A. Corneth_, Mar 28 2021: (Start)
1814400 = 8*9*10*12*14*15 = 15*16*18*20*21 is not in the sequence as the factor 15 is used more than once.
104613949440000 = 12*14*15*16*18*20*21*22*24*25*26 = 20*21*22*24*25*26*27*28*30*32 is not here because, among others, the factor 20 is used more than once.
115880067072000 = 4*6*8*9*10*12*14*15*16*18*20*21*22 = 8*9*10*12*14*15*16*18*20*21*22*24 is not here because most factors are used more than once. (End)
		

Crossrefs

Cf. A002808, A064224, A163263, A342876.

Formula

Numbers of the form Product_{i=x..x+k} A002808(i) = Product_{i=y..y+k-1} A002808(i), where y > x + k and k > 1.

Extensions

Definition edited by N. J. A. Sloane, Apr 18 2010

Keyword:base removed by R. J. Mathar, Apr 24 2010